The Philosopher's Stone has several features and is the base at which an aspiring Alchemist starts. It can be used to craft the Transmutation Tablet, which has many more functions.

Before you can use the stone, you need to charge it. There is nothing special that goes into charging a Philosopher's Stone, just press V until it is fully charged.

The Philospher's Stone not only gives you the power to change a handful of items into others, but it can also transform the very environment around you!
Right Click will transmute certain blocks to others, Shift + Right Click will change a block as well:

Right Click

Shift + Right Click

Stone <=> Cobblestone

Stone ⇒ Grass

Grass <=> Sand

Cobblestone <=> Grass

Dirt ⇒ Sand

Sand ⇒ Cobblestone

Gravel <=> Sandstone

Dirt ⇒ Cobblestone

Water ⇒ Ice

Lava ⇒ Obsidian

Netherrack ⇒ Cobblestone

Logs ⇒ Other logs

Leaves ⇒ Other leaves

Flowers ⇒ Other Flowers

Melon <=> Pumpkin

This effect costs no Fuel. Transmuting large areas at once can incur a cooldown on the effect.

When holding the Philosopher's Stone and pressing C, it allows you to craft in a 3x3 grid just like a crafting table. This is useful in combination with the portable Transmutation Tablet to create anything on the go. However, be cautious, as using the Philosopher's Stone in the crafting grid may cause the stone to pop out of your inventory.

Pressing R will fire a projectile outwards, which, upon hitting a mob, will transform it into another random mob of that disposition (peaceful or aggressive) and type (Overworld or Nether). This effect costs 1 glowstone dust. Mobs added by other mods will turn into a slime or sheep randomly regardless of that mob's disposition. (Chickens are harder to hit, and the best way to turn them is to shoot from a fair distance)